Why the velocity of a wave that propagates through liquid only depends on its medium?

Question

I know that it isn't the case when the wave is light. But like water wave, I learned that the velocity only depends on the medium it propagates, irrelevant with its freq or wavelength. Why it is true?

Answer 1

No and No. The concept of a wavelength dependent propagation speed, $c(\lambda)$, is usually phrased not in terms of speed and wavelength, rather frequency ($\omega = 2\pi c(\lambda)/\lambda$) and wavenumber ($k=2\pi/\lambda$) are used. The relationship between them is called a dispersion relation:

$$ w = c(k) $$

Note that

$$ v_{\rm phase} = \frac{\omega} k $$

is the phase velocity of the wave--that is, the speed of the crests.

$$ v_{\rm group} = \frac{d\omega} {dk} $$

is the group velocity, which is the speed of the envelope (an thus, information).

The 1st "No" was for light propagation. While in a vacuum:

$$ \omega = ck $$

so that all waves propagate at the same speed, the speed of light, and the propagation is dispersionless:

$$ v_{phase} = v_{group} = c $$

this is not the case in a medium. Air has been discussed here (Frequency dependence of the speed of light in air). So "yes", it depends on the medium, but that dependence is wavelength dependent.

The 2nd "No" was for water--now there are many types of water waves like capillary waves, shallow water waves, and gravity waves. Gravity waves in deep water (relative to the wavelength) have the following dispersion relation:

$$ \omega = \sqrt{gk} $$

so that the phase velocity is

$$ c_{phase} = \sqrt{\frac g k} $$

and the group velocity is:

$$ c_{group} = \frac 1 2 \sqrt{\frac g k} = \frac 1 2 c_{phase} $$

So the group velocity is half the phase velocity (going as the square root of wavelength--longer waves move faster, which is what causes the crest of a breaker to overtake the trough) . You can observe also this by throwing a rock in a pond: the crest will appear at the rear of wave packet (ring) and propagate to the front and disappear.

Dispersion relations also play important roles all over the place, e.g: ionosphere delay correction to GPS, seismology, solitons, renormalization, sonar imaging, cosmology, and so on.